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Quantum States and Measures on the Spectral Presheaf

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After a brief introduction to the spectral presheaf, which serves as an analogue of state space in the topos approach to quantum theory, we show that every state ρ of the von Neumann algebra N of physical quantities of a quantum system determines a certain measure μρ on the spectral presheaf of the system. The so-called clopen subobjects of the spectral presheaf play the role of measurable sets. Measures on the spectral presheaf can be characterised abstractly, and the main result is that every abstract measure μ induces a unique state ρμ of the von Neumann algebra N. Finally, we show how quantum-theoretical expectation values can be calculated from measures associated to quantum states.

Document Type: Research Article


Publication date: June 1, 2009

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  • ADVANCED SCIENCE LETTERS is an international peer-reviewed journal with a very wide-ranging coverage, consolidates research activities in all areas of (1) Physical Sciences, (2) Biological Sciences, (3) Mathematical Sciences, (4) Engineering, (5) Computer and Information Sciences, and (6) Geosciences to publish original short communications, full research papers and timely brief (mini) reviews with authors photo and biography encompassing the basic and applied research and current developments in educational aspects of these scientific areas.
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